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\begin{document}

\title{高等代数一}
\subtitle{12-习题与问答-矩阵的运算 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
%\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
\date{{\ppr 2022年11月1日} }

\maketitle

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\begin{frame}{内容提要 }

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\begin{enumerate}

\item  计算矩阵的乘法
\item  计算伴随矩阵与逆阵
\item  求解矩阵方程
\item  化为相抵标准形


\end{enumerate}


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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{讲解本次作业的同学 }

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{\small 
\begin{table}[ht]
\centering
\begin{tabular}{cccccc}
4-习题&8-习题&12-习题&16-习题&20-习题&24-习题 \\ \hline 
{01}&{02}&\underline{03}&04&05&06 \\   
{07}&{08}&\underline{09}&10&11&12 \\  
{13}&{14}&\underline{15}&16&17&18 \\ 
{19}&{20}&\underline{21}&22&23&24 \\  
{25}&{26}&\underline{27}&28&29&30 \\  
{31}&{32}&\underline{33}&34&35&36 \\  
{37}&{38}&\underline{39}&40&41&42 \\  
{43}&{44}&\underline{45}&46&47&48 \\ 
{49}&{50}&\underline{51}&52&53&54 \\  
\end{tabular}
\end{table}
}

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\begin{frame}{12.1. 矩阵的乘法 }

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\begin{itemize}

\item  习题1：计算矩阵的乘积 
{\footnotesize 
\begin{eqnarray*}
(1) \begin{pmatrix} 1&n \\ 0&1 \end{pmatrix}
\begin{pmatrix} 1&m \\ 0&1 \end{pmatrix}, 
\hspace{0.3cm}
(2) \begin{pmatrix} \cos\alpha&-\sin\alpha \\ \sin\alpha&\cos\alpha \end{pmatrix}
\begin{pmatrix} \cos\beta&-\sin\beta \\ \sin\beta&\cos\beta \end{pmatrix}. 
\end{eqnarray*}
}
\item  解答思路：按矩阵乘法的定义进行计算。
{\footnotesize 
\begin{eqnarray*}
(1) \begin{pmatrix} 1&n+m \\ 0&1 \end{pmatrix}, 
\hspace{0.3cm}
(2) \begin{pmatrix} \cos(\alpha+\beta)&-\sin(\alpha+\beta) \\ \sin(\alpha+\beta)&\cos(\alpha+\beta) \end{pmatrix}. 
\end{eqnarray*}
}

\end{itemize}

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\begin{frame}{12.2. 矩阵代入多项式 }

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\begin{itemize}

%Kostrikin#17.5(b)
\item  习题2：设矩阵 {\footnotesize $A=\begin{pmatrix} 2&1&1 \\ 1&2&1 \\ 1&1&2 \end{pmatrix}$}, 设多项式 $f(x)=x^3-3x+2$, 计算 $f(A)$. 

\item  解答思路：
\begin{enumerate}
\item  按定义计算，注意常数项要乘以单位矩阵。
{\footnotesize 
\begin{eqnarray*}
f(A)=\begin{pmatrix} 18&18&18 \\ 18&18&18 \\ 18&18&18 \end{pmatrix}. 
\end{eqnarray*}
}
\item  先将矩阵 $A$ 写成单位矩阵与全1矩阵的和，然后代入多项式。
\end{enumerate}

\end{itemize}

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\begin{frame}{12.3. 矩阵乘法的交换性质 }

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\begin{itemize}

\item  习题3：设矩阵 {\footnotesize $A=\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}$}. 
求出使得 $AB=BA$ 成立的所有矩阵 $B$. 

\item  解答思路：待定系数法。设 {\footnotesize $B=\begin{pmatrix} a&b&c \\ u&v&w \\ x&y&z \end{pmatrix}$}. 
代入等式 $AB=BA$ 求得
{\footnotesize $B=\begin{pmatrix} a&b&c \\ 0&a&b \\ 0&0&a \end{pmatrix}$}, 其中 $a,b,c$ 可取任意实数。 


\end{itemize}

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\begin{itemize}

\item  习题4：设有实数矩阵 {\footnotesize $A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}$}. 设对任意实数矩阵 $B$ 都成立 $AB=BA$. 证明矩阵 $A$ 是一个数量矩阵，即 {\footnotesize $A=\begin{pmatrix} a&0 \\ 0&a \end{pmatrix}$}. 

\item  解答思路：取不同的矩阵 $B$, 例如 
{\footnotesize 
\begin{eqnarray*}
B=\begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix}, 
\begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}, 
\begin{pmatrix} 0&0 \\ 1&0 \end{pmatrix}, 
\begin{pmatrix} 0&0 \\ 0&1 \end{pmatrix}, 
\end{eqnarray*}
}
得到关于矩阵 $A$ 的很多限制条件。

\end{itemize}

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\begin{frame}{12.5. 矩阵乘法的交换性质 }

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\begin{itemize}

\item  习题5：设有两个 $n$ 阶矩阵 $A,B$ 满足条件 $AB=BA$. 证明：
\begin{enumerate}
\item  $(A-B)(A+B)=A^2-B^2$; 
\item  $(A-B)(A^2+AB+B^2)=A^3-B^3$; 
\item  $(A+B)(A^2-AB+B^2)=A^3+B^3$. 
\end{enumerate}

\item  解答思路：按照分配律将左边展开，使用题目所给条件进行化简。注意使用矩阵乘法的结合律。

\end{itemize}

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\begin{frame}{12.6. 对称矩阵与反对称矩阵 }

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\begin{itemize}

\item  习题6：将下述矩阵 $A$ 写成一个对称阵 $B$ （即有 $B^t=B$） 与一个反对称阵 $C$ （即有 $C^t=-C$）的和的形式，
{\footnotesize 
\begin{eqnarray*}
(1)\,\,\, A=\begin{pmatrix} 1&2 \\ 3&4 \end{pmatrix}, 
\hspace{0.3cm}
(2)\,\,\, A=\begin{pmatrix} 1&2&3 \\ 4&5&6 \\ 7&8&9 \end{pmatrix}. 
\end{eqnarray*}
}

\item  解答思路：待定系数法。注意到二阶对称阵和反对称阵具有一般形式
{\footnotesize 
\begin{eqnarray*}
B=\begin{pmatrix} a&b \\ b&c \end{pmatrix}, 
\hspace{0.3cm}
C=\begin{pmatrix} 0&d \\ -d&0 \end{pmatrix}. 
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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\begin{frame}{12.7. 用伴随矩阵和初等变换方法计算逆阵 }

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\begin{itemize}

%Kostrikin#18.9(h)
\item  习题7：设有实数矩阵 {\footnotesize $A=\begin{pmatrix}  3&-4&5 \\ 2&-3&1 \\ 3&-5&-1  \end{pmatrix}$}. 
\begin{enumerate}
\item  计算伴随矩阵 $A^*$ 与行列式的值 $\det(A)$, 并由此写出逆阵 $A^{-1}$. 
\item  使用行初等变换方法计算逆阵 $A^{-1}$. 
\end{enumerate}

\item  答案：{\footnotesize $A^{-1}=\begin{pmatrix}  -8&29&-11 \\  -5&18&-7 \\ 1&-3&1  \end{pmatrix}$}. 

\end{itemize}

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\begin{itemize}
%Kostrikin#18.3(f)
\item  习题8：求解矩阵方程 
{\footnotesize 
\begin{eqnarray*}
\begin{pmatrix} 1&2&-3 \\ 3&2&-4 \\ 2&-1&0  \end{pmatrix}X 
= \begin{pmatrix} 1&-3&0 \\ 10&2&7 \\ 10&7&8  \end{pmatrix}. 
\end{eqnarray*}
}

\item  答案：将 $(A,B)$ 做行初等变换，可得所求矩阵为 {\footnotesize $X=\begin{pmatrix}  6&4&5 \\  2&1&2 \\ 3&3&3  \end{pmatrix}$}.

\end{itemize}

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\begin{itemize}

\item  习题9：将矩阵 {\footnotesize $A=\begin{pmatrix}  1&2 \\ 4&5  \end{pmatrix}$} 写成一些初等矩阵的乘积。

\item  解答思路：先用行初等变换将 $A$ 化为单位矩阵。记录每次初等变换所对应的初等矩阵。

\end{itemize}

\end{frame}

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\begin{itemize}

%Kostrikin#7.1(d)
\item  习题10：用初等变换将下述矩阵化为相抵标准形，写出对应的初等矩阵，
{\footnotesize 
\begin{eqnarray*}
A=\begin{pmatrix} 8&-4&5&5&9 \\ 1&-3&-5&0&-7 \\ 7&-5&1&4&1 \\ 3&-1&3&2&5  \end{pmatrix}
\to 
\begin{pmatrix} E_r & O \\ O & O  \end{pmatrix}. 
\end{eqnarray*}
}

\item  解答思路：先化为行最简形。$R(A)=3$. 这个题目中，对矩阵 $A$ 做行初等变换对应于左乘四阶初等矩阵，做列初等变换对应右乘五阶初等矩阵。

\end{itemize}

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